Differential analyzer



Jan. 10, 1950 s, DARLINGTON 2,494,036

DIFFERENTIAL ANALYZER Filed Sept. 20, 1947 2 Sheets-Sheet 1 FIG.

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lNl 'NmR 5. DA RL/NGTON M w AGENT Jan. 10, 1950 s. DARLINGTON 2,494,035

DIFFERENTIAL ANALYZER Filed Sept. 20 1947 2 Sheets-Sheet 2 lNl/ENTO R 5. DA RL W6 7' ON AGENT Patented Jan. 10, 1950 DIFFERENTIAL ANALYZER Sidney Darlington, Summit, N. J., assignor to Bell Telephone Laboratories, Incorporated, New York, N. Y., a corporation of New York Application September'20, 1947, Serial No. 775,287

This invention relates to an improved electrical difierential analyzer, particularly adapted to obtain the running solution of an equation of the form In this equation, E(t) may be a voltage representing a quantity which is a known function of time, P1 and P2 are expressions involving various powers in the operator while V(t) is a voltage representing a quantity also a function of time and known to be continuously related to the first function as stated in the equation. Illustratively,

may be omitted from the system of the inve ntion, their places being taken by electrical components.

Therefore, another object of the invention is to provide an electrical system for the automatic solution of a differential equation, with constant coeflicients, in a single variable. i

The description which follows is of a preferred embodiment of the invention, andi's to be read with reference to the accompanyin drawings, in

which:

Fig. 1 is a diagram of a circuit'for the solution of an equation of the form 4 Claims. (Cl. 235.61)

of which the symbols have beenabove "defined- 2 Fig. 3 is a circuit diagram similar to that of Fig. 1 but dispensing with the motors there shown.

In all figures, like numerals and letters indi- 1' cate like elements.

It is convenient at the outset to mention the sources of various known devices which occur in some or all of the figures in the accompanying drawings. Reversing amplifiers, identified by the symbol RA, are well known; they include each a vacuum tube in a conventional circuit of unit gain providing an output voltage equal but opposite in polarity to an input voltage. Summing amplifiers, 2A, are the invention of K. D. Swartzel, Jr., disclosed in United States Patent 2,401,779, June 11, 1946. Electrical differentiators, D, are the summing amplifiers of Swartzel modified to provide an output voltage which is the time derivative of the sum of the input voltages; a suitable modification is the invention disclosed in United States Patent 2,412,227, December 10, 1946, to H. G. Och and K. D.

Swartzel, Jr. Alternatively to the differentiators D, one may use an amplifier, conventionally stabilized by a feedback resistor variable for gain adjustment, and apply to the input circuit through a condenser the voltage to be differentiated; such differentiation through a condenser is disclosed by E. S. L. Beale et al. in United States Patent 2,251,973, August 12, 1941. Servomotors, designated S, may be of any convenient type of which many are known; direct current servomotor circuits are shown for example by JA. Rey and by F. A. H. Harl in United States Patents 1,086,729, February 10, 1914 and 1,268,712, June 4, 1918; the shaping of potentiometer cards so that the brush voltage is a desired function of the brush position is disclosed by P. A. Borden in United States Patent 2,114,330, April 19, 1938.

Referring now to Fig. 1, the equation to be dealt with is .21 V- PZE where as above-mentioned may be a quantity varying with time and the P's are expressions involving d 2 d 31% 8? etc. For illustration there is chosen the case where the e'quationiis of the second degree:

aiE l bipE l-oipwfdi=azV+bzpV+cz =V+d2 thei-jcoefficients"areiissumed to be positive.

" -ln-Figsz 1- and 2;-b1 and bz will be represented y directions Amplifier I6 is stabilized vby, reverse as functions of E and of V, defined by the positions, respectively, of brush 2I on potentiometer 2B and of brush on potentiometer 53, which symbolize each an arbitrary function. On linear potentiometer Ill shunting battery II, brush I2 5 is set by shaft 9 so driven by motor M that its angular position continuously represents the quantity E5; -to effect-this; terminals I'and 210i motor M=a-re supp1iedrespectively with a voltage proportional to E, derived in any convenient way and symbolized by generator 3, and withvthe voltage E via conductor 4 from brushJL. Motor M then rotates shaft 9 to keep the voltage between terminals I and 2 zero. Brush IQQis-thus made to derive a voltage to ground-continuouslyfproa portional to E; this voltage is fractionatedpro portinally to m by tap I3:;handset;uon potene tiometer I4, and this fractional voltage aiE is applied through resistor I5 to the input of-sum ming amplifier I6. Through reversing amplifier go I 8:-the:voltage--E is rfediio differentiator I 9 fro which the.-voltage:

Ti'f

islapplied across potentiometer 20,: which .may be non-linear asn shown.- v Onthis potentiometer; brush ZJI ist-also driven by, motor Mzto selecttthe voltage b pfifin which vvthrough resistor? 22 is' sup pliedto the inputof summingt amplifiera' lfis In- 3 1 like; manner pthrough reversing; amplifier; 23; clifierentiatorl 24, potentiometer 25 and-handset tapflfi; the voltage pEiis again differentiated.- and awvoltagercipr Eris obtained and'ssuppliedtothe-input of amplifier lfietl'irongh resistor 21: 5;

Throughwresistor 28,;amplifier I6; receives=-a-voltage .difrom batter 29,l,potentiometer; and handsetztapfillz Theevoltage E'may atany timew beread. on 2 scale; 5 concentric with Upptentiometer III.-

-S'ervomotori32cis supplied from the output-cir cuit oframplifierrl 6 and drives shaft 33 concentricwith circular" potentiometer 35: which shunts" battery-3B; Insulated: from and'carried byshaft 33 rissbrush r3'4-sselectingron potentiometer '3 from 45, battery 36 ayoltage: which is-to bezproportional. to V, theiselution of, the- -equ-ation,- which-:may-ber read ,onseale Thewoltagel at-brush:34 is appliedacross-potentiometer 31 from whichhandset tap llrzappliesrafractional voltage: a2V; through gresistor 39:: to the-input ofampli-fierrl 6... To -thatzinput aresupplied also voltages- -bzpV; -czp V and' 'd2',- through resistors 40;- M and? 42; respectively: Theseaevoltages rare obtained'ftli'rough the operartion of reversingampl-ifiers-45'-and: 48, difieren tiators=-41 and-'48,- and-ibatteryv49y-through tap 50 (the setting of'rwliich' varies with-shaft 33' andso" With-V7, and handset gtamrEkand- 52F-on poten-' v tiometers-5:3, 5.4 andi55 5(of Ewhich :53 "may be nonlinear as shown) in the same manner-asrthe cor: responding functions of E, and the voltage d1. The output connection-from amplifier I6 to motor 32 ,islso made that haft 33is continuously driven to a position where the totaljinputyoltage toame 6O plifier I5 is zero, thereby making the magnitude. of the voltage at brush 34 continuously proportional to V, which maybe read on scale 6 concentric with potentiometer 35. As in all servomotor circuits, for;a given polarity ofqdrivingrvoltage the motor maybe:-conneetedgtogrotaterirr therdesiredi feedback through resistor. 43; i

It is clear-thaw by increasing-' the numberof--- difierentiatorsiand reversing amplifiers irreitl'ir 0 spon-ding tonew coefficient 'rectly impressed across branch, terms of higher order in E or in V may be introduced, and further that both negative and positive values of E and of V are provided for Obvious changes may be made to provide for negative coefiicients, and for other variations of coeflicients with E and V besides the dependence of In on E and b2 on V.

SincetheeflifiirentiatorsioffFi'g. 1 are the summingamplifiers of-Swartzel modified to provide output voltages which are the negative time derivatives of the total respective input voltages, the circuit-of Fig; 1 may be modified to the alternative form shown in Fig. 2, with a considerable economy of- Tapparatu-s Ih'Fig'."2;batteries II and 36 are shunted as in Fig.- l'by'potentiometers I0 and 35 with the respectiveita ps; I2,,a:nd,s34 whereby voltages E and V are obtained. The constant terms (Z1 and dz'are'derived from batteries 29 and 49 and supplied through resistors 28 and 42 to the input of amplifier lfili Motor 32 is controlled by the output vo'ltage of' amplifier Itito' drive shaft 33 car ry-ing-"brusli 34-. Taps I3 and" 38; respectively; supply to amplifier It the input voltages '(ZTE via 25 resistor I5 and a2V via resistor 38.

Potentiometers 20 and 53 of Fi'gkl'are replaced respectivelyby potentiometers 2i! and 53", which differ 'fromthem'only as'tozshaping. The changes in shape produce new potentiometerratios corre;

s b1"; be which are related to b1, b2 by These relations are chosen in: order" that -c2V from taps 5!} and EI, and from these the required time derivatives-are produ'ced'b'y differene ti'a-tors {in and-62 withreversin-g amplifiers El 63fvoltages ciE 'andczV'are fedthrough resistors -65 and B 6-to' diiferentiator "60 from which the output" voltage -c1pE-l-czpV is. taken to ,re-

ence to diffrentiator 62 throughresistor"61.

Voltages 'vi E a'nd i bz' 'V are supplied to diff-j, ferentiator 62 via resistors .68 and 691.,v The total input voltage to difierentiator 162 is thus b1"E+c1pEb2"Vc2pl/' the-w differentiator;output I is then reversed by -re versing amplifier 63 and; viaresistor "It the in- I put circuitwof, amplifier I6 receives the voltage age of amplifier I6 is thus the same as in the. circuit of Fig. 1 :andzthe operation of motor SZ'aga'in equatiomof the ,-second:idegreein an. The necessary additions to deal with equations of higher" deg'reetiin. pcareeobviousi When the coefiicients oittlie'variousspowers o f p are all constants, servomotor 32 may be omitted and the output V ofa summing amplifier I6 di- 'shjowrfinql lgiftiporrespondin or" Iinear=- potentiometer 53 and motor-dri i l m i i 1 e.

r50 are replaced by linear otenaemeiees' and handset tap 50'. Likewise, non-linear potentiometer 20 and motor-driven brush 2| are replaced by linear potentiometer 20' and handset tap 2|. Obviously, the omission of servomotor 32 is equally possible when only the coefficients (12, b2 and 02 are constants, leaving the terms in E to be derived as in Figs. 1 and 2. In either case, the voltage V is read by suitable means, such as voltmeter 90 across potentiometer 31. The input voltage E(t) across resistor M in Fig. 3 corresponds to the voltage E, also a function of time, across resistor M of Fig. 1, and the reading on voltmeter 90 varies with time to show continuously the solution of the equation. Similar modifications in the circuit of Fig. 2 are obviously possible when the coefficients are constant. Coeflicients can also be made to depend on elapsed time, in an arbitrary manner, by using shaped potentiometers with brushes driven by a constant speed motor.

What is claimed is:

1. A system of apparatus for solving a differential equation of the form where E is a given quantity, P1 and P2 are each a polynomial in the operator and V is the solution desired, comprising means for establishing a voltage e1 proportional to E, controllable means for establishing a variable voltage e2, a summing amplifier having an input and an output circuit, means including a plurality of difierentiators for applying to the input circuit voltages individually proportional to the terms of the expression P1e1 together with voltages individually negatively proportional to the terms of the expression Pzez, whereby the voltage of the output circuit is proportional to P1e1+Pz62, and means for applying the output voltage to control the controllable means so to vary the voltage e2 as to reduce the output voltage to the minimum value required to cause the controllable means to maintain e2 negatively proportional to V.

2. Apparatus for the solution of a differential equation of the form in which E is the magnitude of a given quantity, P1 and P2 are each a polynomial in the operator and V is the quantity for which the equation is to be solved, comprising means for establishing a first voltage proportional to E, a first group of differentiating means for successively differentiating the first voltage with respect to time, computing means for deriving voltages proportional to the several terms of HE from the first voltage and its successive differential voltages, a summing amplifier having an input circuit and an output circult, means for combining in the input circuit of the amplifier the voltages corresponding to the terms of PlE, a servomotor driven from the output circuit of the amplifier and controlling potentiometric means for establishing a second voltage, a second group of differentiating means for successively differentiating the second voltage with respect to time, computing means for deriving voltages negatively proportional to the several terms of PzV from the second voltage and its successive differential voltages and means for combining in the input circuit of the amplifier the voltages corresponding to the terms of PzV with those corresponding to the terms of PiE whereby the servomotor is driven to establish by the potentiometric means a voltage negatively proportional to V.

3. Apparatus for the solution of a differential equation of the form P1E=P2V, where E is a given quantity, P1 and P2 are polynomials in the operator and V is the solution desired, comprising a source of voltage, means including electrical differentiators for deriving from the source a sum of voltages individually proportional to the terms of PE, a source of variable voltage, means including electrical diflerentiators for deriving from the second-named source a sum of voltages individually negatively proportional to the terms of PzV, means for comparing the two sums and means controlled by the comparing means for controlling the variable voltage to make numerically equal the sums, whereby the variable voltage is made negatively proportional to V.

4. Apparatus for the solution of a differential equation including a given term independent of the dependent variable in the equation comprising a first and a second source of voltage, means for deriving from the first source a first voltage proportional to the given term, controllable means for deriving from the second source a variable voltage, means including electrical difierentiators for deriving from the variable voltage a plurality of voltages individually proportional to the terms of the equation other than the given term, means for summing the first voltage with the plurality of voltages and means operated by the summing means for driving the controllable means to derive from the second source a voltage such that the net voltage sum becomes substantially zero, whereby the variable voltage is made proportional to the solution in the same ratio as the first voltage is proportional to the given term.

SIDNEY DARLINGTON.

REFERENCES CITED The following references are of record in the file of this patent:

UNITED STATES PATENTS Number Name Date 1,315,539 Carson Sept. 9, 1919 2,233,415 Hull Mar. 4, 1941 2,414,430 Nisbet Jan. 14, 194'! 

